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Search: id:A133079
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| A133079 |
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Expansion of f(q)^3 - 3 * q * f(q^9)^3 in powers of q^3 where f() is a Ramanujan theta function. |
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+0
3
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| 1, -5, -7, 0, 0, 11, 0, -13, 0, 0, 0, 0, 17, 0, 0, 19, 0, 0, 0, 0, 0, 0, -23, 0, 0, 0, 25, 0, 0, 0, 0, 0, 0, 0, 0, -29, 0, 0, 0, 0, -31, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 35, 0, 0, 0, 0, 0, -37, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 41, 0, 0, 0, 0, 0, 0, 43, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -47, 0, 0, 0, 0, 0
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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There is a plus sign which should be minus on each side of Ramanujan's equation.
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REFERENCES
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B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, see p. 357, Entry 5, Eq. (5.1)
S. Ramanujan, Notebooks, Tata Institute of Fundamental Research, Bombay 1957 Vol. 1, see page 266
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FORMULA
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Expansion of f(q) * a(-q) in powers of q where f() is a Ramnaujan theta function and a() is a cubic AGM function.
G.f. is a period 1 Fourier series which satisfies f(-1 / (2304 t)) = -192 (t/i)^(3/2) g(t) where q = exp(2 pi i t) and g(t) is g.f. for A116916.
a(n) = b(24n+1) where b(n) is multiplicative and b(2^e) = b(3^e) = 0^e, b(p^e) = (1+(-1)^e)/2 * p^(e/2) if p == 1, 3 (mod 8), b(p^e) = (1+(-1)^e)/2 * (-p)^(e/2) if p == 5, 7 (mod 8).
a(5*n+3) = a(5*n+4) = 0. a(25*n+1) = -5*a(n).
G.f.: Sum_{k} kronecker(8, 2*k+1) * (6*k+1) * x^(k * (3*k+1)/2).
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EXAMPLE
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q - 5*q^25 - 7*q^49 + 11*q^121 - 13*q^169 + 17*q^289 + 19*q^361 - ...
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PROGRAM
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(PARI) {a(n) = if( issquare( 24*n+1, &n), n * (-1) ^ (n%8 > 4), 0)}
(PARI) {a(n) = local(A, p, e); if( n<0, 0, n = 24*n+1; A = factor(n); prod(k = 1, matsize(A) [1], if(p = A[k, 1], e = A[k, 2]; if(p < 5, 0, p *= kronecker(-8, p); if( e%2, 0, p^(e/2) )))))}
(PARI) {a(n) = local(A); if( n<0, 0, n *= 3; A = x*O(x^n); polcoeff( eta(-x + A)^3 - 3 * x * eta(-x^9 + A)^3, n))}
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CROSSREFS
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(-1)^n * A116916(n) = a(n). A133089(3*n) = a(n).
Sequence in context: A048658 A001111 A116916 this_sequence A080332 A134756 A011350
Adjacent sequences: A133076 A133077 A133078 this_sequence A133080 A133081 A133082
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KEYWORD
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sign
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AUTHOR
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Michael Somos, Sep 08 2007
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